Xu Da and Zhu Yuanzhang played a game of Go. Assuming Xu Da had to defeat Zhu Yuanzhang, what was the probability that Xu Da would use the pieces to form the words "Long Live" on the board? Explain your reasoning.
The Chinese Mathematical Society had a way of challenging the limits of contestants with its problems.
Damn, you needed to know the rules of Go just to do a math problem. How could that not piss you off? Only Chinese students could possibly solve a problem like this. Foreigners would weep after reading it.
This was the National Competition, the toughest high school math contest in all of China.
Any contestant who knew nothing about Go was out of luck. If you didn't have a special skill, how could you even show your face at the National Competition?
Shen Qi knew how to play Go, chess, flying chess, and animal chess. Leaving his skill aside, he knew the rules.
Besides, Shen Qi knew that quite a few of the other National Competition contestants understood the rules of various board games and played them pretty well. For example, the Ebei Province Math Olympiad team could play blindfolded.
"Look past the surface and get to the heart of the matter. This problem has nothing to do with Zhu Yuanzhang or Xu Da. It's just a math problem. Apart from the final question about probability and this Go position, the historical anecdote is all a smokescreen."
Shen Qi quickly thought of Fermat and Pascal's theory on dividing up wagers. In a sense, playing chess was also a form of gambling; people had made a living at it beneath overpasses for ages.
And where there was gambling, there was inevitably probability theory and number theory.
It didn't matter how the words "Long Live" had been formed. They were simply the result of a probabilistic event, a little trick for those who understood math. If Zhu Yuanzhang had known math, he would have punished Xu Da on the spot. And what was the point of giving him Mochou Lake?
The theory of dividing up wagers, jointly formulated by Fermat and Pascal, along with the related theories that followed, provided the mathematical basis for casinos around the world to make a steady profit. Calculating the probability of the words "Long Live" appearing followed principles similar to calculating the odds of getting two kings and four twos, with a straight left in your hand.
Shen Qi began writing: Let the black stones be p and the white stones be q. If p is the probability of a single event occurring, then q is the probability of that event not occurring.
Then, the probability of that event occurring at least m times in n trials is equal to the sum of the terms in the expansion of (p+q) to the nth power, from the p-to-the-n term through the term containing p to the mth power multiplied by q to the (n-m)th power.
Using this theory, Shen Qi quickly calculated the probability of the words "Long Live" appearing: just 0.02 percent. He also explained his reasoning in detail.
Calculating probabilities wasn't hard. Once you understood the math above, you could become a gambling king too. The hard part was walking out of a casino alive, with all your limbs intact.
Shen Qi reasoned that Zhu Yuanzhang and Xu Da really had played Go, but the story about Xu Da forming the words "Long Live" to win Mochou Lake was most likely just a legend.
Mathematically speaking, the words "Long Live" would appear only once in every fifty thousand games. A game of Go could take anywhere from dozens of minutes to several hours, and playing three to five games a day was already a lot.
If Zhu Yuanzhang and Xu Da did nothing but play Go every day, they would have to play for twenty-seven years before seeing "Long Live" formed by the black and white stones.
But Zhu Yuanzhang was the founding emperor. Didn't he have state affairs to deal with?
Of course, it was possible that the "Long Live" event happened on the very first of those fifty thousand tries.
So it was just a legend. It couldn't be taken seriously.
"This first problem, huh? At first glance, it's a real pain in the ass, but once you've solved it, the pain's gone—and you might even feel a little shiver of pleasure. It's actually pretty interesting. Ah, I've never been to Mochou Lake. I'd love to see it." Shen Qi ate a Snickers bar to celebrate solving the National Competition's first problem.
Without wasting a moment, Shen Qi moved on to the second problem, which dealt with plane analytic geometry.
For a Level 5 mathematician like Shen Qi, analyzing affine transformations of two-dimensional homogeneous coordinates with determinants wasn't difficult. It was just a matter of finding a set of invariants for rotation, translation, and reflection.
The idea that single elliptic geometry corresponded to a subgroup of projective transformations might seem like an obvious axiom, but he couldn't be fooled by appearances. That would only lead him astray.
The most levelheaded Math Olympiad contestants would go straight to the heart of the matter and find the imaginary absolute conic in the plane. The second problem was then practically a gift.
Shen Qi used an economical, practical method to find the imaginary conic. The Klein continuous transformation taught in university textbooks was far too complicated; it was just making trouble for yourself.
Poincare, hailed as the world's last "universal scholar," was clearly more flexible. Shen Qi loved using Poincare's many ideas and conclusions.
From the provincial contest to the national contest, Shen Qi had used Poincare's theories to solve problems more than once. Poincare was a polymath in mathematics and a master of physics, astronomy, philosophy, and other fields as well.
There were many ways to obtain an absolute conic from a curve in the coordinate plane. Poincare's degenerate coincidence method was practically a magic weapon for math competitions. Shen Qi used that very method: direct, simple, and decisive. It worked incredibly well, as if it had been made for math competitions.
Two hours had passed, and Shen Qi had solved two problems. He drank some Dongpeng Special Drink to perk himself up, along with teddy bear biscuits, a Snickers bar, and Wife Cake to replenish his energy.
Don't think math was purely mental work. It took physical stamina too. Just sitting there and writing nonstop—try doing it for two hours and see how tired you got.
The sixty National Competition contestants were spread across seven classrooms. There were ten contestants in Shen Qi's room, each from a different province or city.
There were as many as eleven proctors, one assigned to watch each contestant, with the last one patrolling.
After all, this was the National Competition. A gold medal in the National Mathematics Competition carried enormous prestige. Once the competition had weeded out the rest, only six people would win gold. Those six lucky students would be snapped up by all the top universities.
The proctor assigned to watch Shen Qi came up behind him and asked softly, "Have you had enough to eat, kid?"
"Yep, I'm full. I'll have another spicy strip for good luck—may things go swimmingly." With a spicy strip in his left hand and a compass in his right, Shen Qi ate as he drew a pair of concentric circles.
"Why did you bring so many snacks into the exam room?" the proctor asked quietly, frowning.
Shen Qi was puzzled. "Huh? Aren't we allowed to bring food? Four and a half hours, sir! It's practically like the imperial examinations in ancient times—a real endurance test. I'll get hungry if I don't eat."
"Just don't eat too much. You'll get sleepy." The proctor kindly reminded him, then took a quick look at Shen Qi's exam paper. From Nanyue Province's Math Olympiad team—Shen Qi. Not bad, this kid. He'd finished two problems in just two hours and was leisurely eating spicy strips. He could be a dark horse!
The proctor watching Shen Qi worked for the Chinese Mathematical Society and was a full-time mathematical researcher himself. He paced to the classroom door and asked a younger man, "How many points did Contestant 56, Shen Qi, get in the National Preliminary?"
The younger man had an iPad. He pulled up the information and glanced at it. "Shen Qi. Twenty-one points in the National Preliminary, a perfect score. It's not easy for a contestant from Nanyue Province to make it to the National Competition by the skin of his teeth and still get a perfect score. Shen Qi's the only one on their team with a perfect score."
"Mm, he's talented. Our society's guiding principle is to recruit talent without sticking to convention." The older proctor nodded, then said to the young man, "Xiao Wen, go buy me a pack of spicy strips, the same kind Shen Qi brought. It's past eleven and I'm hungry."
(Heh heh heh, everyone, toss me some recommendation votes! (*▽*))
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